Posted By: riazi
The Probabilistic Method has recently been developed intensively and became one of the most powerful and widely used tools applied in Combinatorics. One of
the major reasons for this rapid development is the important role of randomness in Theoretical Computer Science, a field which is recently the source of many intriguing
combinatorial problems.
The interplay between Discrete Mathematics and Computer Science suggests an algorithmic point of view in the study of the Probabilistic Method in Combinatorics
and this is the approach we tried to adopt in this book. The manuscript thus includes a discussion of algorithmic techniques together with a study of the classical method as
well as the modern tools applied in it. The first part of the book contains a description of the tools applied in probabilistic arguments, including the basic techniques that
use expectation and variance, as well as the more recent applications of Martingales and Correlation Inequalities. The second part includes a study of various topics in
which probabilistic techniques have been successful. This part contains chapters on discrepancy and random graphs, as well as on several areas in Theoretical Computer
Science; Circuit Complexity , Computational Geometry, and Derandomization of randomized algorithms. Scattered between the chapters are gems described under
the heading "The Probabilistic Lens". These are elegant proofs that are not necessarily related to the chapters after which they appear and can be usually read separately.
The basic Probabilistic Method can be described as follows: in order to prove the existence of a combinatorial structure with certain properties, we construct an
appropriate probability space and show that a randomly chosen element in this space has the desired properties with positive probability. This method has been initiated
by Paul Erdos, who contributed so much to its development over the last fifty years, that it seems appropriate to call it "The Erdos Method", His contribution cannot be
measured only by his numerous deep results in the subject, but also by his many intriguing problems and conjectures that stimulated a big portion of the research in
the area.